Research the ChurchTuring thesis State what the thesis is wh

Research the Church-Turing thesis. State what the thesis is, what its importance with respect to computation generally and with respect to the material in the book Godel, Escher, Bach.

Solution

It states that a function on the natural numbers is computable by a human being following an algorithm, ignoring resource limitations, if and only if it is computable by a Turing machine.

Church-Turing thesis concerns the notion of an effective ormechanical method in logic and mathematics. ‘Effective’ and its synonym ‘mechanical’ are terms of art in these disciplines: they do not carry their everyday meaning. A method, or procedure, M, for achieving some desired result is called ‘effective’ or ‘mechanical’ just in case

A well-known example of an effective method is the truth table test for tautologousness. In practice, of course, this test is unworkable for formulae containing a large number of propositional variables, but in principle one could apply it successfully to any formula of the propositional calculus, given sufficient time, tenacity, paper, and pencils.

Statements that there is an effective method for achieving such-and-such a result are commonly expressed by saying that there is an effective method for obtaining the values of such-and-such a mathematical function. For example, that there is an effective method for determining whether or not any given formula of the propositional calculus is a tautology -- e.g. the truth table method -- is expressed in function-speak by saying that there is an effective method for obtaining the values of a function, call it T, whose domain is the set of formulae of the propositional calculus and whose value for any given formula x, written T(x), is 1 or 0 according to whether x is, or is not, a tautology.